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Data Analysis: Measures of Dispersion
Once we understand the measures of central tendency,
and check them in our data, we are then concerned
with how spread out (or dispersed) the values are. A
statistic that conveys this information is a measure of
dispersion.
Maybe the simplest measure of dispersion is the range,
which is the difference between the greatest and the
smallest value for a given variable, among the n observations.
It is a simple concept often used, but it does not tell
us anything about the distribution of the values of the
variable of interest, whether they are concentrated
around the middle, spread out all over, or concentrated near the endpoints.
It is also very sensitive to outliers Think of comparing the following samples
a) 2, 4, 4, 4, 4, 6
b) 2, 2, 2, 6, 6, 6
c) 2, 2, 2, 6, 6, 9
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A modified measured is the interquartile range: it is
the result of the following operation
3rd quartile value 1st quartile value
(1)
that means it is the range of the middle 50% of observations. It is the difference between the 75th percentile and the 25th percentile.
But we can do better. And for that we need a simple concept, the deviation of the values of a variable
with respect to its mean. For the ith observation, the
deviation of X from the mean is
di Xi X (2)
since there are n observations, there are n deviations,
some of them are positive and some of them are negative.
We can try to summarize these deviations and compute for example the mean deviation. However, the
positive and negative values of the deviations always
cancel out, so the mean deviation always equal zero:
n
di ∑n
i 1
∑
Xi X n
(3)
but of course we can rewrite that as
Xi
∑n n
X nX X
0
∑n
n
i 1
2
(4)
To overcome this problem we can develop other measures, like the Mean Absolute Deviation. But other
are more useful for us, like for example squaring the
deviations, calculating their mean and then taking
the square root. It seems like a lot of steps, but it
is so commonly used that it is worth the effort:
SX ∑ni 1 di2 n 1
∑ni
Xi X n 1
1
2
(5)
Notice that we are now dividing by n 1 instead of
n, this is due to the fact that once we know X,i and
n 1 values of a variable, the nth observations is determined. Here n 1 is what we call the degrees of
freedom.
The Standard Deviation is usually interpreted as a
typical deviation, but it is better to remember that
usually 68% of the observations have values within
one standard deviation of the mean, that is in the
interval X SX X SX . Also about 95% of the
observations have values within two standard devia
tions of the mean, that is in the interval X 2 SX X 2 SX .
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Another typical statistic is the variance, which is defined as the square of the standard deviation:
SX2
∑ni 1 di2 n 1
∑ni
Xi X 2
n 1
1
(6)
Obviously, the standard deviation and the variance
carry the same information, we will use both depending on the context. Again, we have to remember
that both measures are sensitive to extreme values.
Notice that the numerator in the definition of the
variance is usually called, total variation in X, as it
serves as a measure of the overall variability among
the values of X.
Finally, it is important to mention that although elegant the formulae for the standard deviation and
the variance are not very practical to compute by
hand, instead it is better to have the following formula for computing the variance (the standard deviation is just its square root):
∑X
∑X2 N
N 1
2
SX2 4
(7)
Data Analysis: Symmetry
Sometimes it is useful to have a measure of the symmetry or asymmetry of a distribution of values. For
that we have the skewness of a distribution defined
as
Xi X 3
∑
µ3
(8)
n 2
Notice that a symmetric distribution has skewness
equal to zero.
A distribution can have positive skew, where the longer
tail is to the right, or a negative skew, when the tail is
to the left. In the case of positive skew the mode and
the median are lower values than the mean. When
the skew is negative the mean is lower than the median and the mode.
Bivariate Statistics
Our ultimate objective in econometrics is to determine how some variables are related to others, here
we take a step in that direction.
We start with a data set that has n observations on
two variables, X and Y .
We could plot these observations, and observe whether
high values of one variable are associated with low
values of the other, or with high values, etc.
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One measure that quantifies how much the values of
X and Y vary together is the covariance:
SXY
1
n
n 1 i∑
1
Xi X Yi Y (9)
Notice that X and Y play the same role here, so there
is no difference between the covariance of X and
Y , and that of Y and X. The covariance only captures linear relationships, and it is very sensitive to
the units used.
To overcome the used of different units we can compute the correlation coefficient that is defined as:
r
SXY
SX SY (10)
The interpretation of the correlation coefficient relates to an imaginary line that can be fit through the
data points. The sign of r is the same as the sign
of the slope of that line, and the magnitude (in absolute value) measures the degree to which th points
lie close to the line.
If the points in a scatter diagram lie exactly along
some straight line with a positive slope, then r 1.
If they lie exactly along a line with a negative slope
then r 1. If the points are scattered all over randomly, the correlation will be close to zero. Notice
that if they lie along a vertical or horizontal line, then
r will not be defined because either SX or SY will be
zero.
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